{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:MIBZMGML57IIPT3TDZKEY56AOV","short_pith_number":"pith:MIBZMGML","schema_version":"1.0","canonical_sha256":"620396198befd087cf731e544c77c0755b607a7014b38eeb340032875043f4b6","source":{"kind":"arxiv","id":"1210.4150","version":2},"attestation_state":"computed","paper":{"title":"New methods to bound the critical probability in fractal percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Henk Don","submitted_at":"2012-10-15T19:57:40Z","abstract_excerpt":"Fractal percolation has been introduced by Mandelbrot in 1974. We study the two-dimensional case, with integer subdivision index M and survival probability p. It is well known that there exists a non-trivial critical value p_c(M) such that a.s. the largest connected component in the limiting set K is a point for p<p_c(M) and with positive probability there is a connected component intersecting opposite sides of the unit square for p\\geq p_c(M).\n  For all M\\geq 2, the value of p_c(M) is unknown. In this paper we present ideas to find lower and upper bounds, significantly sharper than those alre"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1210.4150","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-10-15T19:57:40Z","cross_cats_sorted":[],"title_canon_sha256":"91093acfe356ba6ef0aeb3ee6fa5747f948264bc34d5d091c2ac11e8792e1009","abstract_canon_sha256":"0d8e4fd42e8ce77830caa546cae6da0bc346a6cfa85d5a1178a6b0cce1257a29"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:06.124839Z","signature_b64":"hgI3SKE4t+J/dS+B7Adgc2FkUx0HLC12MZnTY3xcfTNBq0EJgTbuS7WPaC/Wy0/adRroYMqT75q1Psm1glBpBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"620396198befd087cf731e544c77c0755b607a7014b38eeb340032875043f4b6","last_reissued_at":"2026-05-18T01:04:06.124418Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:06.124418Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New methods to bound the critical probability in fractal percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Henk Don","submitted_at":"2012-10-15T19:57:40Z","abstract_excerpt":"Fractal percolation has been introduced by Mandelbrot in 1974. We study the two-dimensional case, with integer subdivision index M and survival probability p. It is well known that there exists a non-trivial critical value p_c(M) such that a.s. the largest connected component in the limiting set K is a point for p<p_c(M) and with positive probability there is a connected component intersecting opposite sides of the unit square for p\\geq p_c(M).\n  For all M\\geq 2, the value of p_c(M) is unknown. In this paper we present ideas to find lower and upper bounds, significantly sharper than those alre"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.4150","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1210.4150","created_at":"2026-05-18T01:04:06.124478+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.4150v2","created_at":"2026-05-18T01:04:06.124478+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.4150","created_at":"2026-05-18T01:04:06.124478+00:00"},{"alias_kind":"pith_short_12","alias_value":"MIBZMGML57II","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_16","alias_value":"MIBZMGML57IIPT3T","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_8","alias_value":"MIBZMGML","created_at":"2026-05-18T12:27:14.488303+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MIBZMGML57IIPT3TDZKEY56AOV","json":"https://pith.science/pith/MIBZMGML57IIPT3TDZKEY56AOV.json","graph_json":"https://pith.science/api/pith-number/MIBZMGML57IIPT3TDZKEY56AOV/graph.json","events_json":"https://pith.science/api/pith-number/MIBZMGML57IIPT3TDZKEY56AOV/events.json","paper":"https://pith.science/paper/MIBZMGML"},"agent_actions":{"view_html":"https://pith.science/pith/MIBZMGML57IIPT3TDZKEY56AOV","download_json":"https://pith.science/pith/MIBZMGML57IIPT3TDZKEY56AOV.json","view_paper":"https://pith.science/paper/MIBZMGML","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.4150&json=true","fetch_graph":"https://pith.science/api/pith-number/MIBZMGML57IIPT3TDZKEY56AOV/graph.json","fetch_events":"https://pith.science/api/pith-number/MIBZMGML57IIPT3TDZKEY56AOV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MIBZMGML57IIPT3TDZKEY56AOV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MIBZMGML57IIPT3TDZKEY56AOV/action/storage_attestation","attest_author":"https://pith.science/pith/MIBZMGML57IIPT3TDZKEY56AOV/action/author_attestation","sign_citation":"https://pith.science/pith/MIBZMGML57IIPT3TDZKEY56AOV/action/citation_signature","submit_replication":"https://pith.science/pith/MIBZMGML57IIPT3TDZKEY56AOV/action/replication_record"}},"created_at":"2026-05-18T01:04:06.124478+00:00","updated_at":"2026-05-18T01:04:06.124478+00:00"}